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Numerical simulation of thermal-hydrologic-mechanical-chemical processesin deformable, fractured porous media

Joshua Taron a,�, Derek Elsworth a, Ki-Bok Min b

a Department of Energy and Mineral Engineering and Center for Geomechanics, Geofluids, and Geohazards (G3), Pennsylvania State University, University Park, Pennsylvania, USAb School of Civil, Environmental, and Mining Engineering, The University of Adelaide, SA, Australia

a r t i c l e i n f o

Article history:

Received 7 August 2007

Received in revised form

28 October 2008

Accepted 20 January 2009Available online 25 February 2009

Keywords:

THMC

Geothermal simulation

CO2

Fracture reactive transport

Reservoir permeability

Dual porosity

a b s t r a c t

A method is introduced to couple the thermal (T), hydrologic (H), and chemical precipitation/dissolution

(C) capabilities of TOUGHREACT with the mechanical (M) framework of FLAC3D to examine THMC

processes in deformable, fractured porous media. The combined influence of stress-driven asperity

dissolution, thermal-hydro-mechanical asperity compaction/dilation, and mineral precipitation/

dissolution alter the permeability of fractures during thermal, hydraulic, and chemical stimulation.

Fracture and matrix are mechanically linked through linear, dual-porosity poroelasticity. Stress-

dissolution effects are driven by augmented effective stresses incrementally defined at steady state with

feedbacks to the transport system as a mass source, and to the mechanical system as an equivalent

chemical strain. Porosity, permeability, stiffness, and chemical composition may be spatially

heterogeneous and evolve with local temperature, effective stress and chemical potential. Changes in

total stress generate undrained fluid pressure increments which are passed from the mechanical

analysis to the transport logic with a correction to enforce conservation of fluid mass. Analytical

comparisons confirm the ability of the model to represent the rapid, undrained response of the fluid-

mechanical system to mechanical loading. We then focus on a full thermal loading/unloading cycle

of a constrained fractured mass and follow irreversible alteration in in-situ stress and permeability

resulting from both mechanical and chemical effects. A subsequent paper [Taron J, Elsworth D.

Thermal-hydrologic-mechanical-chemical processes in the evolution of engineered geothermal

reservoirs. Int J Rock Mech Min Sci 2009; this issue, doi:10.1016/j.ijrmms.2009.01.007] follows the

evolution of mechanical and transport properties in an EGS reservoir, and outlines in greater detail the

strength of coupling between THMC mechanisms.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

It is well known that fractured rocks exhibit changes inmechanical compliance and hydraulic conductivity when sub-jected to thermal, hydraulic, mechanical, and chemical forces. Inmany engineering applications it is important to be able to predictthe direction and magnitude of these changes. However, theinterplay between temperature, effective stress, chemical poten-tial, and fracture response is complex: it is not only influencedby anisotropic and spatially varying fracture properties, but alsoby fracture properties that are dynamic, and evolve with thedynamic nature of the applied forces.

The gaping or sealing of natural fractures has clear implica-tions in reservoirs for the sequestration of CO2 [2] and radioactivewaste repositories [3], where the release of CO2 or the redistribu-tion of pore fluids around contained radioactive waste is a primary

concern. Volcanic environments are also impacted, as in the caseof failing volcanic domes [4], where elevated fluid pressuresmay destabilize an existing volcanic pile. In other cases, suchas petroleum or gas reservoirs, hot dry rock [5] or enhancedgeothermal systems [6] (HDR/EGS), engineered stimulations maybeneficially improve fluid circulation; a topic of significantinterest since the majority of worldwide geothermal capacity iscontained within low permeability rock masses [6,7].

Despite their importance, the competing influence of processesthat degrade fluid conductivity in dominant fractures, such asthin-film pressure solution [8–10] and mineral precipitation, andthose that enhance it, such as shear dilation [11,12], mineraldissolution [13–15], and strain energy driven free-face dissolution[9,16] have yet to be addressed at geologic scale. To examine theseprocesses together, a link between chemical and mechanicalbehavior that maintains dependence on thermal and hydrologicchanges is required, i.e., THMC coupling. And while severalTHM [3,17–20] and THC [14] coupling methodologies have beensuggested, to the authors’ knowledge no single numericalsimulator has been introduced to examine THMC processes in a

ARTICLE IN PRESS

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/ijrmms

International Journal ofRock Mechanics & Mining Sciences

1365-1609/$ - see front matter & 2009 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijrmms.2009.01.008

� Corresponding author. Tel.: +1814 863 9733; fax: +1814 865 3248.

E-mail address: [email protected] (J. Taron).

International Journal of Rock Mechanics & Mining Sciences 46 (2009) 842–854


construct that is applicable to the broad variety of above-mentioned engineering applications.

Fig. 1 illustrates the potential error in excluding the chemi-cal–mechanical link from numerical modeling. In the figure, wefollow a complete cycle of thermal/stress loading in a chemicallyactive fractured rock. During the loading/unloading cycle, rever-sible (elastic) and irreversible (chemical–mechanical: pressuresolution or other) changes in aperture occur, with the ultimateresult that after unloading, once the system has been returned toits initial background state, we see an irreversible aperturereduction, and a corresponding irreversible loss in the state ofstress. These two occurrences (7 and 8 in Fig. 1) are the behaviorsof primary interest, as they indicate a complete and potentiallysignificant alteration of the resting system that cannot berepresented without the inclusion of THMC processes.

2. Model capabilities

We now introduce and implement a method for coupling themultiphase, multi-component, non-isothermal thermodynamics,reactive transport, and chemical precipitation/dissolution capa-bilities of TOUGHREACT [14] with the mechanical frameworkof FLAC3D [21] to generate a coupled THMC simulator. This‘‘modular’’ approach, first proposed by Settari [22] to couple

geomechanics with reservoir flow simulation, has some advan-tages over the development of a single coupled program. Modularapproaches will typically be more rapid and less expensive todevelop, although working within the framework of an existingcode can sometimes lack the freedom that is inherent in ‘‘fromscratch’’ code development. Additionally, as pointed out by Settariand Mourits [23], the modular construction allows for easierimplementation of future advances in constitutive relationshipsor modeling structures (rather than modifying an entire codingstructure), and the system can utilize highly sophisticated,rigorously validated existing codes developed at high cost. It cantake many years for a new modeling structure to be validated bythe research community, but in the case of TOUGHREACT andFLAC3D, each has been extensively scrutinized and each code is‘‘qualified’’ for regulated programs, such as the US radioactivewaste program.

Furthermore, single codes often simplify behavior beyond theprincipal scope of the analysis. For example, complex geomecha-nical codes may represent the flow system as only single phase,and complex reactive transport codes often incorporate mechan-ical response as invariant total stresses. Appropriate couplingenables the important subtleties of geomechanical response to befollowed while maintaining complex fluid thermodynamics andreactive processes. Although development time is shortened inthis modular approach, execution times are commonly extended,as neither code is optimized for the couplings, and data transfermust occur between the concurrently or sequentially executingcodes. As suggested by Settari and Mourits [23] and Minkoff et al.[24], however, this may not always be the case, because in systemswhere geomechanics may be loosely coupled (not changing at arapid pace) the geomechanics simulation may not need to beconducted very often, thus improving computational efficiencyover fully coupled codes where mechanics are equilibrated atevery fluid flow time step.

The coupled analysis that we present incorporates featuresunique to engineered geosystems, particularly those underelevated temperature and chemical potential, involving theundrained pressure response in a dual-porosity medium andstress-chemistry effects including the role of mechanicallymediated chemical dissolution of bridging fracture asperities.FLAC3D is exercised purely in mechanical mode, where undrainedfluid pressures may be evaluated (externally) from local totalstresses. This undrained methodology allows calculation of theshort-term build-up in fluid pressures that result from aninstantaneous change in stress, provided we have knowledge ofthe compressibility of the pore fluids and the solid matrix. In thisway, the complex thermodynamics of phase equilibria of multi-phase water mixtures, and even multi-component mixtures (suchas CO2 and water), can be tracked in the pre-existing framework ofTOUGHREACT. As TOUGHREACT has no use for compressibility,however, it is necessary to code this capability into the programor, as we have done, to insert a thermodynamic calculation intothe external linking module (discussed later). For water mixtures,we utilize the 1997 International Association for the Propertiesof Water and Steam (IAPWS) steam table equations [25]. For CO2

mixtures, an appropriate equation of state would be required, andwe have not yet added this capability. If a system is unsaturated(such as in HDR/EGS), fluid compressibility is very large, and theundrained poroelastic equations approach their drained counter-parts. Therefore, while our construct is tailored to saturatedsystems, drained systems are automatically accommodated.

FLAC3D is applied independent of time to accommodatethe incremental equilibration of stresses for various mechanicalconstitutive relationships. TOUGHREACT performs time-dependenttransport calculations, tracking thermodynamic relationshipsfor temperature, phase equilibria, and pore pressure dissipation

ARTICLE IN PRESS

Fig. 1. Conceptual, behavioral trend of thermally loaded and fractured rock: (A)

follow light gray line as (1) increasing temperature builds stress (partially reduced

by elastic fracture strain). (2) Irreversible fracture strains reduce stress, which, for

illustrative purpose, is applied at the end of loading (3). Thermal unloading follows

the black line. (B) Follow gray temperature (stress) loading line (4) elastic

reduction in fracture aperture (idealized as linear). Loading reaches maximum

value (5). Aperture irreversibly closes (chemical strain) and causes corresponding

drop in stress. Black (6) unloading line returns the system to its resting state for an

(7) irreversible aperture reduction and (8) corresponding irreversible stress loss.

J. Taron et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 842–854 843


together with aqueous chemical equilibrium and kinetic pre-cipitation/dissolution in a dual-porosity medium. Under largethermal stresses, shear failure may be expected, and FLAC3D iscapable of handling this with the constitutive theories ofMohr–Coulomb or Hoek–Brown. Plastic flow is also possible,although this would require consideration of permeabilitychanges that occur during fracture shear and also fracturecompression. This complexity is not addressed here, and will bethe topic of a future paper.

3. Simulation logic

Simulation is executed within FLAC3D’s FISH programminglanguage [21], where external operations by TOUGHREACT andthe linking module are controlled. TOUGHREACT, an integral finitedifference code [26], calculates all properties at the centralcoordinate of element volumes. In contrast, the first order finitedifference program FLAC3D, with explicit temporal derivatives anda mixed discretization method that overlays constant strain-ratetetrahedral elements with the final zone elements (adding greaterfreedom in methods of plastic flow), utilizes properties of state(p,T) at corner nodes and mechanical variables (s,u) at centralcoordinates. Correspondingly, state properties from centralTOUGHREACT nodes are interpolated to connecting corner nodesof FLAC3D. Stress (not displacement) outputs from FLAC3D are usedas the independent variable in constitutive relationships. Theparsing of stresses to TOUGHREACT is direct, as they are calculatedcentrally within the node-centered blocks of FLAC3D (in spatialagreement with TOUGHREACT).

In its current construction, the codes iterate upon the samenumerical grid. This structure, however, is not required. Aspointed out by Minkoff et al. [24], un-matched meshes are onebenefit to a modular code. For example [24], it may be desirable toconduct flow simulations upon a reservoir area impacted by fluidinjection and withdrawal only, while the mechanical grid mayinclude the reservoir area in addition to all overburden up to theground surface. Neither must the overlapping simulation areasutilize identical grid spacing, such that it may be desirable torefine the fluid flow mesh to capture some complex physics in aspecific area, without adapting the mechanical mesh to agree. It isonly required that interpolation of data accommodate thedifferences in mesh extent and geometry.

Sequential execution of the two programs is linked by aseparate code capable of parsing data outputs from each primarysimulator as input to the companion. This separate code isreferred to as the ‘‘interpolation module’’. The module is a Fortran90 executable, and maintains access to data outputs fromTOUGHREACT and FLAC3D. In addition to data interpolation, thismodule executes constitutive relationships including permeabil-ity evolution, dual-porosity poroelastic response to stress, andthermodynamically controlled fluid compressibility.

All transient calculations take place within TOUGHREACT, andit is here that the time step is controlled for conditions of fluidvelocity, grid size, and reaction rates. Additionally, there is asecondary (explicit) time step that controls how often stress iscorrected to changes in fluid pressure (for what length of timeTOUGHREACT conducts a flow simulation before allowing stressequilibration in FLAC3D). This frequency is controlled in theinterpolation module. If the magnitude of stress change in thesystem over one time step is beneath a pre-determined tolerance,the frequency is decreased (if stress is not changing, mechanicalre-equilibration is unnecessary), and vice versa for an uppertolerance. Coupling is explicit and constitutive calculations areperformed once per iteration (assuming constant constitutivevalues throughout a fluid flow time step), requiring sufficiently

small time steps relative to the rapidity of change in the system.The validity of utilizing such a methodology is discussed in latersections to provide insight into this explicit time step.

The coupling cycle is shown in Fig. 2, and is comparable to theloose coupling, modular structure of Minkoff et al. [24] andRutqvist et al. [3]. Simulation begins with equilibration oftemperature (T) and pore fluid pressure (pf) in TOUGHREACT,where porosity (f) changes due to mineral precipitation/dissolu-tion and liquid saturation (S) are also obtained. Constitutiverelationships in the interpolation module transform these outputsinto fluid bulk modulus (Kf), as obtained from IAPWS steam tableequations, and permeability change due to mineral behavior(DkTC). The TOUGHREACT central node data (pf,T) are theninterpolated to corner node information as input to FLAC3D. Afterstress equilibration in FLAC3D, the interpolation module usesstress outputs within a dual-porosity framework, consisting ofmatrix (pf

(1)) and fracture (pf(2)) pore fluid pressures, to obtain the

pressure response to the new stress field via domain (matrix,fracture) and state (p,T) specific Skempton coefficients. Effectivestress is then used to obtain the permeability change due topressure solution type behavior (DkTMC) while chemical strain (eC)is accommodated in the stress field (discussed later). Parametersthen re-enter TOUGHREACT for the next time step.

4. Governing equations

The physical system of interest is modeled herein as a multi-continuum, fully or partially saturated fracture/matrix systemwith direct communication between the domains. Local thermalequilibrium is assumed between the fluid and solid (at a singlepoint in continuum space, the fluid and solid exhibit the sametemperature), but not between separate fracture and matrixdomains. From this framework, a differential of pressure andtemperature may develop between the fracture and matrix, withproperties of pressure and temperature dissipation influencingthe rapidity of transfer from local changes in the fracture systeminto the surrounding matrix blocks, and vice versa. As such,the multi-continuum distinction is fully maintained withinthe numerically represented THC system, while local continuityof stress requires equilibrium of stresses between fracture andmatrix, which is then represented within the single continuumframework of FLAC3D. For this transition, physical characteristics

ARTICLE IN PRESS

Fig. 2. Coupling relationship between TOUGHREACT, FLAC3D, and the interpolation

module.

J. Taron et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 842–854844


are delegated based upon dual-porosity poroelastic theory[27–31]. The governing balance equations and their constitutivecounterparts are discussed below.

4.1. Conservation of momentum—solid

Mechanical equilibrium of the solid phase is governed by thebalance of linear momentum,

sij;j þ bi ¼ r _vi, (1)

where bi are the body forces per unit volume, _vi are the materialtime derivatives of velocities, and sij,j represents the divergence ofthe transpose of the Cauchy stress tensor. In an iterativeformulation, for static equilibrium of the medium, the momentumbalance becomes the common force equilibrium relation

sij;j ¼ �bi. (2)

The resulting unknowns can be related to each other throughany of several elastic or plastic constitutive relationships. Webegin with the case of an isotropic, elastic solid, thus introducingthe stress/strain constitutive relationship for a medium with twodistinct porosities (see dual-porosity discussion below), includingthe effects of pore fluid pressure, p, and temperature, T (acombined equation utilizing constitutive poroelasticity (e.g. [32],Eq. 7.42), with thermoelastic response, and utilizing two distinctpore fluid pressures as in Wilson and Aifantis [27]),

sij ¼ 2G�ij þ2Gn

1� 2n �kkdij � ðað1Þp pþ að2Þp pÞdij � aT Tdij, (3)

where G is the shear modulus, v is the Poisson ratio, aðiÞp and aT arethe coupling coefficients for fluid and thermal effects for the (1)

fracture and (2) matrix, dij is the Kronecker delta, and thelinearized (‘‘small’’) strains are defined as the symmetric part ofthe displacement gradient ui,j, i.e.,

�ij ¼12ðui;j þ uj;iÞ. (4)

Inserting Eq. (4) into Eq. (3) and the result into the equilibriumequation, Eq. (2), yields the Navier equation for the displacements,u

Grui þG

1� 2nuk;ki ¼ ðað1Þp p;i þ að2Þp p;iÞ þ aT T ;i � bi, (5)

4.2. Conservation of momentum, mass, and energy—fluid

Fluid, aqueous species, and energy are transported through thesystem as defined by their respective mass and energy balances.The master equation for these processes is given in integral formas

d

dt

ZV

MkdV ¼

ZG

Fk � nþ

ZV

qkdV , (6)

where the left-hand side represents the rate of accumulation ofthe conserved quantity (Mk is mass of fluid, mineral mass, orenergy density) resulting from the arrival of the fluxes Fk, (of fluid,mass, or energy) across the boundary, G, and complemented byvolume sources, qk, distributed over the nominal element volume,V, for each component, k (gas, liquid, advected species, or heat). Inthis discussion we have adopted (for clarity of coefficients)standard tensor notation, where bold values represent first orsecond order tensors. Eq. (6) may be transformed into its commonPDE counterpart through use of the divergence theorem

qMk

qt¼ �r � Fk þ qk, (7)

where the mass, flux, and source terms must then be indepen-dently determined for a given system.

Mass, or energy density, Mk, in Eq. (7) is defined for eachcomponent, k, as the summation of the various contributions tothe component across all phases (subscripted l, g, s for liquid, gas,or solid) as

Mk ¼ fSlrlXl þ fSgrgXg þ ð1� fÞrsXs, (8)

where S is phase saturation, r is density (or species concentra-tion), f is porosity, and Xs,l,g is mass fraction (or internal energy).Simplification then occurs for each calculation. The third termdisappears for fluid mass calculations (no solid phase present),while the second and third terms are excluded from aqueousspecies mass (species may be present within the liquid medium,but not solid or gaseous).

Fluxes, F, in Eq. (7) are given by the summation across phases(b ¼ l,g) of the advective and diffusive terms as

F ¼Xb¼l;g

�Xbrbkkr

b

mbðrpb � rbgÞ

!� lbrC, (9)

where the first term represents the contribution of advectionthrough consideration of the multiphase extension of Darcy’s lawfor relative permeability, kr, intrinsic permeability vector, k,dynamic viscosity, m, and, as before, r is density of fluid (orconcentration of species) and X is mass fraction for fluid transport,specific enthalpy for heat flow, or unity for chemical calculations.The second term represents diffusive transport as governed by thelaws of Fick and Fourier, and introduces conductivity, lb, andgradient (of temperature or concentration), rC. This last diffusiveterm is only present when calculating the flux of temperature orconcentration, and therefore disappears when calculating pureliquid flux. For heat flow calculations, lb is thermal conductivity,while for chemical flux lb ¼ rbtfSbDb with tortuosity, t, anddiffusion coefficient, Db. Note that a hydrodynamic dispersionconcept is not utilized in the classic Fickian sense. Instead,TOUGHREACT utilizes the interaction of regions with differingvelocities (fracture and matrix in a dual-porosity construct) toinduce solute mixing [33]. In the case of mineral mass, the fluxterm disappears (colloid transport is not considered).

The source term, qk, in Eq. (7) may be comprised of an injectionor withdrawal source or as an increase in species concentration(or mineral mass) due to dissolution (or precipitation). A thermalsource may also arise due to a release of energy during chemicalreactions. This last case is not currently considered. Sources ofaqueous species and/or mineral mass are discussed in thefollowing.

4.3. Chemical precipitation/dissolution

A generalized rate law for precipitation/dissolution of amineral, m, is [34,35],

rm ¼ sgnðlogðQm=KemÞÞk

cmAmf ðaiÞ 1�

Qm

Kem

� �j��n

, (10)

where kc is the rate constant, A is the specific mineral reactivesurface area per kg of H2O, Ke is the mineral/water equilibriumconstant, and Q is the ion activity product. The function f(ai)represents some (inhibiting or catalyzing) dependence on theactivities of individual ions in solution such as H+ and OH� [36],and sgnðQm=Ke

mÞ provides a direction of reaction: positive forsupersaturated precipitation. The exponential parameters, j andn, indicate an experimental order of reaction, commonly assumedto be unity. An additional term (multiplied by Eq. (10)) may alsobe introduced to represent the dependency of reactive surfacearea on liquid saturation [33]. Dependency of the rate constant

ARTICLE IN PRESS



may be handled, to a reasonable approximation [37], via theArrhenius expression,

kc¼ kc

25 exp �Ea

Ru

1

T�

1

298:15

� �� , (11)

for the rate constant at 25 1C, k25, activation energy, Ea, and gasconstant, Ru.

In the case of amorphous silica an alternate expression may beused following [38], where the precipitation rates reported in [39]were observed to underestimate behavior in geothermal systems.This new rate law, based upon experimental data for morecomplex geothermal fluids, becomes, in a form modified in [40] toapproach zero as Q/K approaches one (i.e., as the systemapproaches equilibrium)

rm ¼ sgnðlogðQm=KemÞÞkmAmf ðaiÞ

Qm

Kem

� �m�

Kem

Qm

� �2m��

��n

. (12)

These are the formulations utilized in TOUGHREACT. Reactionsbetween aqueous species (homogeneous reactions) are assumedto be at local equilibrium, and therefore governed by therelationship between the concentrations of basis (primary)species and their activities, partitioned by the stoichiometriccoefficients. This relationship is termed the law of mass action(e.g. [34]). The assumption of local equilibrium greatly reduces thenumber of chemical unknowns and ODEs (between primary andsecondary species), and is accurate to the extent that the truereaction rates outpace the rate of fluid transport in a given system.This is a correct assumption for most aqueous species [34] (andflow systems), but less so for slower redox reactions [33,34]. InTOUGHREACT, species activities are obtained from an extendedDebye–Huckel equation with parameters taken from [41].

5. Deformable dual-porosity material

To represent the pressure loading of a fully or nearly liquidsaturated system (particularly at high temperature and pressureand with multi-component liquids) coupling of the aboveformulation requires the undrained (instantaneous) response ofpore fluid pressure to mechanical loading in both the fracture andmatrix domains. Hydrologic considerations allow a timed pres-sure-dissipation response throughout the fracture dominatedfluid system and between the fracture/matrix companionshipfollowing undrained loading.

Classically, a dual-porosity material is represented as a porousmatrix partitioned into blocks by a mutually orthogonal fracturenetwork [42,43]. In this scenario, permeability is much higherwithin the fracture network, thus allowing global flow to occurprimarily through the fractures, while the vast majority of storageoccurs within the higher porosity matrix (due to its larger globalfraction of the medium). Interchange of fluid and heat betweenfractures and matrix, so-called ‘‘interporosity flow’’, is driven bypressure or temperature gradients between the two domains.

Expansion of this classic two-domain interaction into ‘‘multi-ple interacting continua’’ [44,45] allows the gradual evolution ofgradients between fracture and matrix through the existence ofone or more intermediate continua placed, mathematically, somelinear distance from the fracture domain. This development hasallowed for numerical approximations to more accurately repre-sent the slow invasion of locally (to the fracture) altered pressuresand temperatures deeply into the matrix blocks, and introduceddispersive mixing that arises at the interface of zones withdiffering fluid velocities. While this multi-continuum methodol-ogy may be adopted in TOUGHREACT to represent dual-perme-ability fluid transport with uniformly constant stress fields in

time, we do not seek such an expansion with respect to a flow-deformation response [46]. As such, a dual-porosity frameworkwith two interacting continua (fracture and matrix) is utilized inthis study, while a compatible poroelastic theory carries thisbehavior into the mechanical domain.

5.1. Fluid pressure response

Extension of Biot’s poroelastic theory [47–50] to a dual-porosity framework has been previously addressed [27–31,46,51]. The methodologies presented in these works provide anadequate framework for the phenomenological representation ofporoelastic coefficients capable of describing flow-deformationresponse in such a medium.

Continuity of fluid mass is represented in a compressiblemedia as,

qzqtþr � F ¼ 0, (13)

where z is the increment of fluid content as in [52], and comprisesthe relative motion between fluid and solid. Inserting Darcy’s lawfor the flux term yields

qzqt�

k

mr2p ¼ q. (14)

Biot’s [48] linear-poroelastic constitutive equivalence, for volu-metric strain, e, is

e

z

!¼

1

K

1

H1

H

1

R

0BB@

1CCA s

p

!, (15)

where the coefficients 1/K, 1/H, and 1/R are the bulk drainedcompressibility, poroelastic expansion, and specific storage,respectively. Substituting

B � �dp

ds

��z¼0

¼R

H, (16)

for the Skempton coefficient, a � K=H, for the Biot–Williscoefficient and

1

R�dzdp

��s¼0

¼a

KB, (17)

for the specific storage, condensing Eq. (15) to relate fluid contentto strain, and substituting its time derivative in Eq. (14),establishes the flow condition for a single-porosity medium withno fluid sources

aBKu

_pþ a_e ¼ k

mr2p, (18)

where we have utilized the relationship for undrained bulkmodulus,

Ku �dsde

��z¼0

¼K

1� aB. (19)

Extending to a dual-porosity medium, we follow the sameprocedure leading to the dual-porosity form of Eq. (5), whereEq. (18) is modified to exhibit two separate fluid pressures(for fracture and matrix) with flow between them governed by, inits simplest form, an instantaneous pressure differential, Dp ¼

(p1�p2) [42], to obtain two continuity relationships [28],

kðiÞ

m r2pðiÞ ¼

aðiÞ

K ðiÞu BðiÞ_pðiÞ þ aðiÞ _eþ ð�1ÞigDp, (20)

where i is not a repetitive index, but represents the existence oftwo separate equations for the matrix (i ¼ 1) and fracture (i ¼ 2),

ARTICLE IN PRESS



and g is the cross coupling coefficient for flow exchange betweenthe two domains [53]. Eq. (20) states that the divergence of fluidflux for a given control volume must equal the rate of accumula-tion within that volume, and is thus a statement of massconservation.

5.2. Dual-porosity load response

The general linear relation between strain, increment of fluidcontent, total stress (s), and pore fluid pressure (p), simplyextends Eq. (15) to allow, again, for two separate fluid pressures[31]

de

�dzð1Þ

�dzð2Þ

0B@

1CA ¼

c11c12c13

c21c22c23

c31c32c33

0B@

1CA

�ds�dpð1Þ

�dpð2Þ

0B@

1CA, (21)

where the superscripts refer to the (1) matrix and (2) fracturedomains. The single porosity coefficients of Biot are no longerapplicable, and are replaced by the uknown coupling coefficients,cij, that may be designated via a phenomenological deconstructionsimilar to that of Biot and Willis [52]. The coefficient matrix canbe shown to be symmetric [31] by the Betti reciprocal theorem.Performing manipulations of the above equation through isolationof independent components (i.e. long-time versus short-timelimits) allows determination of the central coefficients (seedetailed procedure in [31,30]).

Herein we assume that c23 ¼ c32 ¼ 0 [31], which differs slightlyfrom the procedure of [27,28,30]. Examination of Eq. (21) showsthat this assumption implies the following: an undrainedapplication of stress that influences a change in fluid content forthe fracture domain does so through modification of fracture fluidpressure, and does not influence that of the matrix. The reverse isalso true, with the overall implication being, see discussion inBerryman and Wang [31], that in the undrained limit the matrixand fracture domains are completely separate. This can beconsidered a justification for a dual-porosity approach [31].

In our analysis, the purpose of dual-porosity elasticity is toattain Skempton coefficients representing both the fracture andmatrix domains

dpð1Þ ¼ Bð1Þds ¼ � c12

c22ds

dpð2Þ ¼ Bð2Þds ¼ � c13

c33ds, (22)

which represent the undrained (dz ¼ 0) build in pore fluidpressure in each domain for a given change in stress as providedby FLAC3D. Relationships to calculate these two Skemptoncoefficients are provided in Table 2 of [31]. For this procedure,we choose as the known coefficients K(1), K, Ks

(1), and Kf, where Ks

is the solid grain modulus (in a microhomogeneous medium [54])and the fluid bulk modulus,

1

Kf�

1

V

dV

dp

��T

, (23)

is calculated in the interpolation module as a function of position,temperature, and pressure utilizing the IAPWS steam tableequations [25]. For a complete reconstruction of the individualrelations required to represent the dual-porosity poroelasticresponse, refer to [31,51].

5.3. Effect on the global mass balance

Injection of fluid mass into TOUGHREACT in the form of fluidpressure violates conservation of mass by an amount proportionalto the compressibility of the local fluids. A change in pressure by

this procedure necessitates a change in local fluid volume, andtherefore appearing or disappearing mass. However, when thelocal element is fully saturated, a stiff fluid will not significantlyrespond (volumetrically) to stress induced pressure changes,while for unsaturated media even a significant volumetricresponse will not in general dictate a noticeable change in mass.Nonetheless, we err on the side of safety and correct for thisdiscrepancy with a recast of Eq. (23),

dV ¼1

KfVdp

��T

, (24)

which indicates the volume (or mass) error due to an increase inpressure, dp (at a given temperature). To correct for potential massloss, we alter elemental volumes (physically reduce the volumeof the mesh element) within TOUGHREACT by this amount (in anintegral finite difference formulation, this does not require thealteration of geometric coordinates). In our simulations, includingboth single and multiphase flow with water/steam phase changesoccurring, we have not detected total system mass losses greaterthan �0.01% of total system mass.

6. Undrained fluid/mechanical response

We now examine the error that our formulation introduces tothe fluid-mechanical coupling. Excluding constitutive approxima-tions, error may be introduced into the coupling procedure as ithas been described up to this point in two primary ways: explicittime step size, and the equilibration step between a stress changeand its undrained pressure response (Fig. 3).

The first is a direct byproduct of explicit coupling, insomuchas an increase in time step (length of the TOUGHREACT fluidstep between each mechanical equilibration), allows a greateramount of fluid pressure to diffuse between each mechanicalequilibration, introducing error proportional to the fluid diffusiv-ity and inversely proportional to the rate of mechanical change(not the amount of mechanical change per timestep, ds, whichimplies proportionality to error, but the rate of change per unittime (ds/dt), implying inverse proportionality).

The second form of error, shown in Fig. 3, is due to the natureof the undrained pressure response, which may not be fullyaccommodated by a single stress equilibrium step. In other words,at a given time step a fixed pressure field enters FLAC3D andis accommodated by a calculated stress distribution. This stressdistribution induces a modification of the previously fixedpressure field, and this new pressure field may, in turn, producea redistribution of the stress field whether or not any fluid isallowed to diffuse (within TOUGHREACT). A number of steps may

ARTICLE IN PRESS

Fig. 3. Relationship between coupling methodologies. Interior looping may occur

over n steps (at fixed time, t ¼ tk) to equilibrate the response of stress to an

undrained increase in pressure. Alternatively, this inter-looping may be excluded

in favor of a ‘‘leapfrog’’ method, where a single stress equilibration (run of FLAC3D)

is conducted per time step.



be required to find the true equilibrium magnitude of stress andpressure, which tends to asymptote at a value higher than issuggested by a single equilibration step. This is not necessarily aMandel-Cryer type effect [55,56], which is a real occurrence andwould require the action of a diffusing fluid pressure andredistribution of stresses around the diffusing magnitudes(although the behavior is comparable). The case where FLAC3D isrun once per explicit time step (single equilibration step) isreferred to herein as the ‘‘leapfrog method’’ (Fig. 3). Each of thesepossible error sources (1explicit time step and 2leapfrog versuspÐ s iteration) requires further examination, which conse-quently leads to validation of the undrained fluid-mechanicalcoupling.

6.1. Fluid-mechanical couple: instantaneous loading

In one dimension, we may examine the accuracy of the fluid-mechanical coupling in comparison to the classical fluid diffusionequation of hydrogeology (e.g. [57]),

qp

qt� cf

q2p

qz2¼ 0, (25)

which is a specific poroelastic result of Eqs. (14) and (15)restricted to a one-dimensional column of soil (or rock) underconstant applied vertical stress [58], and gives its form to theanalytical solution for heat flow [59, p. 96]

pðz; tÞ ¼4p0

pX1m¼0

1

2mþ 1expð�c2cf tÞ sinðczÞ, (26)

where C ¼ ð2mþ 1Þp=2L, and p0 ¼ B(v)s0 is the initial undrainedpressure response to the applied vertical stress (s0). The one-dimensional Skempton coefficient (‘‘loading efficiency’’ in [58]) isgiven by

BðvÞ ¼ �Bð1þ vuÞ

3ð1� vuÞ, (27)

for the Skempton coefficient, B, and undrained Poisson ratio, vu.This is the canonical consolidation problem of a one-dimensionalcolumn of soil subjected to a constant vertical stress applied att ¼ 0+ to the top of the column, with fluid pressure allowed todrain freely from the point of applied stress. A similar solution isavailable for column displacement u (e.g. [48,58]),

q2u

qz2¼ cm

qp

qz, (28)

for Geertsma’s [60] uniaxial expansion coefficient (consolidationcoefficient), cm � a=K ðvÞ, with uniaxial bulk modulus, K(v)

¼ K+4G/3.Under the same boundary conditions as above, the analyticalsolution is [58]

Duðz; tÞ ¼ cmp0 ðL� zÞ �8L

p2

X1m¼0

1

ð2mþ 1Þ2expð�C2cf tÞ cosðCzÞ

" #,

(29)

with definitions the same as for Eq. (26), and the instantaneousdisplacement at the time of stress application u(z,0+) ¼ s0(L�z)/Ku

(v), for the undrained unaxial bulk modulus,

K ðvÞu ¼Kuð1þ vuÞ

3ð1� vuÞ. (30)

All undrained parameters approach their drained counterparts asfluid compressibility becomes large, or fluid saturation ap-proaches zero.

Results of a TOUGHREACT-FLAC3D simulation mimicking theseboundary conditions are presented against these analyticalsolutions in Fig. 4. A column of porous rock (E ¼ 13 GPa,

v ¼ 0.22) with displacements constrained laterally and porepressure initially zero, is subjected to an applied vertical load,s0 ¼ 50 MPa, at t ¼ 0+, and pressure is allowed to drain freely fromthe top of the column only. Time step was chosen large enough toillustrate the error incorporated in very early times (near the timeof undrained loading) due to the leapfrog method of simulation(Fig. 3).

Pressure builds up (and elastic displacement decreases) in theearly stages as the model cycles between stress equilibrationand undrained pressure response (leapfrog artifact). Following theinstantaneous loading period (50 MPa applied over one time step)numerical results overlay nearly identically the analytical solutionas pressure diffuses and stress accommodates the pressurereduction. A slightly greater error occurs at points nearest thefree draining surface (left-most curve in Fig. 4A) due to the explicittime step size, where a greater rate of fluid diffusion allows thefluid to move greater distances before being accommodated by amechanical response.

6.2. Fluid-mechanical couple: constant loading rate

In light of Fig. 4, it is of interest to examine more precisely theerror that arises while the sample is being loaded. To do so, wewish to utilize the same geometry, but apply the load graduallyover a finite loading period at a given loading rate, ds0/dt (rate of

ARTICLE IN PRESS

Fig. 4. Comparison of TOUGHREACT-FLAC3D fluid-mechanical coupling simulation

versus analytical results in one dimension: (A) normalized (p0 ¼ p(z,0) ¼ B(v)s0)

pressure diffusion response versus diffusive time (tD ¼ ct/L2); (B) normalized

(u1 ¼ uð0;1Þ ¼ s0L=K ðvÞ) displacement response versus diffusive time.



increase of applied load at the top of the column per unit time).Here, we maintain the one-dimensional form, but alter thegoverning diffusion equation (25) to accommodate a constantloading rate [58]:

1

cf

qp

qt�q2p

qz2¼

cmmk

ds0

dt, (31)

with the series solution adjusted so that, as above, the freedraining boundary is at z ¼ 0 [59, p. 130],

pðz; tÞ ¼cmds0

dt

L2m2k

1�ðL� zÞ2

L2�

32

p3

X1m¼0

ð�1Þm

ð2mþ 1Þ3

� expð�C2cf tÞ cosðCðL� zÞÞ

!. (32)

Results of the gradual loading analysis are presented in Fig. 5.Loading rate refers to the rate of increase of applied load at the topof the column per unit time. The amount of load change periteration (ds0) is a function of the time step (dt), so that a smallerload change is experienced per iteration as the time step isdecreased. Time steps were chosen for A and B such thatds0 ¼ (ds0/dt)�dt is the same magnitude in each case. Fromthe figure, two primary conclusions are apparent. Firstly, atthe slowest loading rate (Fig. 5A) and smallest time step(and correspondingly smallest value of ds0) there is no differencebetween the leapfrog approach and a simulation with additionalpÐ s iteration (inter-looping), proving the intuitive result thatsmall explicit time steps remove the need for inter-looping. In this

case, if the time step is too large to capture the fluid-mechanicalcoupling, then inter-looping has little effect because more error isintroduced by the fluid-mechanical couple than by the leapfrogmethod (evidenced by the fact that the dashed lines do notimprove in accuracy over their corresponding solid lines).Secondly, a faster loading rate (Fig. 5B) results in greater errordue to the leapfrog method, but lesser error due to the explicittime step size (evidenced by the relative accuracy of all threedashed lines). In other words, mechanical change (loading) isfaster relative to fluid diffusion, and so the explicit time step sizemay be larger and still accommodate the fluid-mechanicalcoupling because less frequent mechanical equilibration isrequired to keep up with the relatively slower fluid diffusion.However, precisely because the loading rate is faster, greater errorwill result due to the non-iterative equilibration of stress andpressure. Therefore, a larger time step is viable, but only withinter-looping. In any case, the system may be accuratelyrepresented with the proper selection of time step and iterativemethod for a given rate of mechanical change, and at the slowerloading rate (likely closer to those that might be seen in naturalsystems) the leapfrog method is sufficient provided that theexplicit time step is reasonably small. For now, experimentation isrequired to guarantee accurate coupling.

7. THMC mediated aperture/permeability change

Having now examined the fluid-mechanical mechanism, weproceed to introduce further complexities that surround chemicalbehavior. And, because constitutive behavior in a geologicalsystem is generally non-linear, responses mediated by stress,fluid pressure, temperature, and chemical potential often requireempirical examination. Notably, permeability of the systemmay change by orders of magnitude in response to changesin effective stress. In the following, we describe changes inpermeability resulting from both stress and chemical effects,utilizing the empirical relationship proposed in [61]. Thatrelationship is further developed herein to accommodateunloading of fracture asperities in a manner that suggestsfracture gaping may occur only through mechanical means(or by thermal contribution to the stress field). Section 7.1presents the governing loading equations as found in [61],whereas Section 7.2 illustrates an unloading construct similar tothat used in [61], but where unloading is allowed to occur onlythrough mechanical means.

7.1. Loading behavior

Hydraulic aperture of a fracture under an applied effectivestress, s0, may be defined empirically as [3]

bm ¼ brm þ ðb

0� br

mÞ expð�os0Þ, (33)

where bm is the hydraulic aperture (subscripted m indicatingchanges due solely to mechanical effects), b0 is the aperture underno mechanical stress, br

m is the residual aperture at maximummechanical loading and o is a constant that defines the non-linear stiffness of the fracture.

The dissolution of bridging asperities may also reduce theeffective aperture of the fracture. These ‘‘chemical’’ effects may beaccommodated in the relationship for fracture aperture in a formthat includes the mechanical compaction process of Eq. (33) andpressure solution-type dissolution of contacting asperities, wherewe have substituted bmax

m ¼ b0� br

m as the maximum possiblemechanical closure [61]

bmc ¼ brm þ fb

rm � br

c þ bmaxm expð�os0Þg � expð�s0ðb� w=TÞÞ, (34)

ARTICLE IN PRESS

Fig. 5. Error (compared to analytical solution) in undrained pore pressure

response for constant loading rate of one-dimensional vertical column for (A)

slower loading rate (ds0/dt ¼ 5.0�104) and (B) faster loading rate (ds0/

dt ¼ 5.0�105). ‘‘Leapfrog’’ method of simulation is solid gray line with data

points. Additional inter-looping method is dashed black line.



where T is temperature and the empirical constants b andw define the chemical compaction process. The subscripted c

represents changes due to chemical effects and brc is the

residual aperture at maximum chemical loading. Buried withinthese two constants (see [61]) is the critical stress [61, modifiedfrom 62,63],

sc ¼Emð1� T=TmÞ

4Vm, (35)

where Em is the heat of fusion, Tm is the temperature of fusion, andVm is the molar volume of the mineral comprising the fractureasperity. Dissolution of the contacting asperity will progresswhere the local asperity stress exceeds this critical stress thatrepresents both the chemical and mechanical potential of thecontact.

Permeability is evaluated for an orthogonal set of persistentfractures of spacing s, from the cubic law, [64,65] k ¼ b3/12 s. Notethat Eq. (34) represents equilibrium behavior, where chemicallymediated changes have run to completion (i.e., it is a thermo-dynamic, not a kinetic relationship).

7.2. Unloading behavior

The above constitutive relationship governs aperture closureunder conditions of thermal/mechanical loading due to the effectsof mechanical deformation (Eq. (33)) and chemical alterationincluding mechanical deformation (Eq. (34)). If utilized in itsentirety and without memory of any previous mechanical/thermalstate, this represents the case of complete reversibility. However,aperture closure should not be viewed as completely reversible orirreversible, but as a mechanism that is dependent on the initialstress state and subsequent loading, as well as one that maintainsmemory of some attained stress magnitude and a subsequentunloading period.

For instance, subsurface storage of radioactive waste ischaracterized by a loading period during which temperaturesteadily increases and fracture apertures correspondingly de-crease, followed by a period of sustained cooling towards thebackground state, implying a reversal of this process (fracturegaping). Alternatively, geothermal reservoirs are largely charac-terized by unloading behavior, where the maximum stress/temperature condition is the in-situ state of the fractured mass,and the injection of cooler circulation fluids causes unloadingfrom this in-situ state. It is of some interest to determine theprecise behavior of such an unloading period and its beginningtransition.

The mechanical component of fracture closure is nota completely reversible process, but exhibits hysteresis asgoverned by both the elastic and plastic properties ofthe contacting asperities. Furthermore, while chemical behaviorsmay contribute to permeability increases through the actionof thermodynamically governed dissolution, pressure solutiontype mechanisms as discussed above are incapable ofinducing gaping of the fracture during an unloading stage(barring the inclusion of ‘‘force of crystallization’’ processes,pressure solution is irreversible). Therefore, it is apparentthat an additional term is needed to describe the reversibleportion of mechanical closure, while excluding the possibilityof chemical reversibility. In this aim, we follow a proceduresimilar to that of Min et al. [61] to develop an unloadingrelationship, but maintain a reversibility that is due purely tomechanical effects.

In the simplest formulation, this need may be addressedthrough a mechanical recovery ratio, Rm, that governs the degreeof elastic reversibility, and is defined as the ratio of the potentialunloading mechanical aperture change, bmax

mðuÞ, to the maximum

potential loading mechanical aperture change, bmaxm , as

Rm ¼bmax

mðuÞ

bmaxm

. (36)

It is first necessary to examine the case of a mass unloaded from astate of infinite stress with the unloading version of Eq. (33)

bmðuÞ ¼ brm þ bmax

mðuÞ expð�os0Þ, (37)

or, from the definition of recovery ratio

bmðuÞ ¼ brm þ Rmbmax

m expð�os0Þ. (38)

However, the unloading process is dependent on the maximumloading stress (initial unloading stress). The difference in aperturebetween this maximum loading stress and some unloaded state is,utilizing Eq. (38),

DbmðuÞ ¼ Rmbmaxm expð�osðuÞ

0Þ � Rmbmaxm expð�osmax

0Þ, (39)

with the maximum (prior to unloading) effective stresssmax

04s(u)0, for any subsequent unloading effective stress, s(u)

0.This inequality states that load cycling is not considered. Theunloaded aperture is then comprised of the difference betweenthis change and the fully loaded aperture, bf:

bmðuÞ ¼ bfþDbmðuÞ. (40)

In the case of mechanical loading and unloading, the aperture atmaximum loading stress, bf, is equivalent to the final loadedaperture, bm(smax

0), and so the unloading aperture is obtained bysubstituting Eq. (33) in Eq. (40). However, we are seeking therelationship for a fracture that has been chemically and mechani-cally loaded, and then unloaded along a path defined by therecoverable portion of mechanical loading. Therefore, substitutingbf¼ bmc(smax

0) and inserting Eq. (39) into Eq. (40) and simplifyingyields

bmðuÞ ¼ bmcðsmax0ÞÞ þ Rmbmax

m fexpð�osðuÞ0Þ � expð�osmax

0Þg, (41)

where bmc(smax0) is Eq. (34) evaluated at s0 ¼ smax

0. This relation-ship defines the aperture at a stress magnitude lower than andobtained a posteriori the fully loaded state. Eqs. (34) and (41) thenfully define the loading and unloading cycle, respectively, of afractured mass. The required empirical parameters are shown inTable 1. Parameters were obtained through a comparison withexperimental results introduced in the heated block test of TerraTek [66], where the aperture was monitored during a completeloading and unloading cycle in-situ, on a 2�2 m cube of graniticgneiss subjected to stresses supplied by flatjacks with tempera-ture alteration via borehole heaters. The original experimentalresults of Hardin et al. [66] are shown in Fig. 6, alongsidetheoretical reproduction of this behavior calculated with Eqs. (34)and (41). In the figure, loading begins at point 9 (and is isothermalfor the first three data points) and continues until point 16(non-isothermally), before being unloaded to the initial state atpoint 21. Hardin et al. also performed two intermediate load/unload cycles at points 13 and 16. These two intermediate cyclesare not considered here, and the analytical solution is incapable of

ARTICLE IN PRESS

Table 1Parameters of the permeability constitutive relationship as utilized in Fig. 6.

Parameter Fitted value

Residual mechanical aperture, brm (mm) 6.0

Residual chemical aperture, brc (mm) 3.0

Constant in aperture relationship, b 1.00

Constant in aperture relationship, w 345

Stiffness coefficient (1/MPa) 0.375

Mechanical recovery ratio, Rm 0.8



representing them. Agreement between the two data sets issatisfactory for the primary points of interest (intermediateloading/unloading cycling is not considered), excluding point 19,where unloading aperture cannot be reproduced with the givenanalytical model (which is purely mechanical and does notundergo unloading with decreasing temperature unless the stressfield is altered).

7.3. Force of crystallization

Chemical precipitation is commonly assumed to cause areduction in fracture aperture due to a buildup of depositedspecies along the fracture face. Contrary to this assumption is theconcept of ‘‘force of crystallization’’, dating back to 1896 with thework of Dunn [67] and 1920 with Tabor [68], with a phenomen-ological model presented by Weyl [8]. Force of crystallizationoperates analogously and inversely to pressure solution where,instead of relieving fracture stress through dissolution at asperitycontacts, if the fluid is sufficiently super saturated mineralprecipitation and crystal growth may exert pressure at contactpoints and lead to physical gaping of the fracture. Furtherdiscussion of the mechanism is available in the literature (e.g.[69–71]). While we do not, in a fundamental sense, implicitlyconsider the impact of this process in our model, the current logicis capable of accommodating this effect in a straightforwardmanner—should solution concentrations be sufficiently supersaturated. The phenomenological relationship for pressure solu-tion that we utilize is able to adequately match the laboratorystudies on which it is based, all of which involve significantlyunder saturated fluids only.

8. THC mediated porosity/permeability change

Thermo–chemical induced changes in permeability may bereferenced to precipitation/dissolution behaviors along the con-tinuum fracture and matrix domains. Here, aperture changes are

caused by the addition or removal of mineral components fromthe walls of (at the scale of these investigations) an assumeduniform fracture face, or an isotropic porous volume fraction. Thisis not precisely ‘‘free face dissolution’’ (which implies contributionof strain energy to thermodynamic dissolution), but a purelychemically driven process governed by the rates of reaction aspreviously discussed. In the following, we assume that processesof this type may act independently from pressure solution over asingle time step, thus enabling them to be additive over that timestep. This does not indicate process independence, which wouldallow chemical analyses to be conducted separately of TM or ofTMC without loss of accuracy. These processes are still stronglydependent on one another outside of a time step. For example,changes in permeability from pressure solution (or chemicalprecipitation/dissolution) will alter the flow characteristics andresidence times of circulating fluids, thus modifying thermaltransport. Changes in local temperature in this manner alter thestress field and modify chemical reaction rates. Modified reactionrates and residence times influence the characteristics of chemicalreaction, while modified temperature and stress influencepressure solution and thermal gaping.

Changes in fracture aperture due to THC behavior areaccommodated via the chemical precipitation behavior incorpo-rated in TOUGHREACT. Addition or removal of mineral mass fromthe continuum system results in a change in fracture or matrixporosity within a nominal element volume, as given by the overallchange in the volume of minerals present by [40,72],

f ¼ 1�XN

m¼1

f rxm � f u, (42)

where frx is the volume fraction of mineral m in the surroundingrock vmineral/vmedium, and fu is the volume fraction of the non-reactive surrounding rock. Relations between fracture porosityand permeability are provided in the literature. One suchpossibility is a simple cubic relationship [34]:

k ¼ kiffi

� �3

, (43)

where the subscript, i, refers to an initial property and k, and f arepermeability and porosity, respectively. While several suchrelations may be implemented from within TOUGHREACT, it isnecessary in our case to calculate permeability changes externallyin order to operate multiple mechanisms simultaneously. Com-patibility between the permeability change due to this behaviorand that of pressure solution can be indexed to the change infracture aperture by, as before, b ¼

ffiffiffiffiffiffiffiffiffiffiffi12ks3p

. Aperture change viathis mechanism is then assumed additive to the THMC aperturereductions associated with pressure solution driven compaction.

Several options also exist for the relationship between matrixporosity and permeability. One such possibility is the Carman–Kozeny equation [73],

k ¼ kið1� fiÞ

2

ð1� fÞ2ffi

� �3

, (44)

where all parameters are as previously defined, although matrixpermeability is likely an insignificant contributor (in many cases)to overall system behavior.

9. Chemical strain and stress

Modifications in fracture aperture necessarily lead to changesin the local stress field. However, because FLAC3D uses grid pointdisplacements to calculate strains, see Eq. (4), and does not storevalues of strain, no provision is available to input strains due to

ARTICLE IN PRESS

Fig. 6. Comparison of the analytical results of Eqs. (34) and (41) against

experimental results of [66]. Experimental results are shown as black dashed line

with solid data points. Gray solid line with hollow data points is the analytical

solution. Each data point is numbered to correspond with the original data points

of [66].



aperture change and subsequently convert them into gridpointdisplacements. An alternative method is needed.

For equally spaced orthogonal fractures, the impact of a changein aperture on local linear strain (unidirectional from a singlefracture) is represented by

�CH ¼Db

s, (45)

where s is fracture spacing and the subscript CH refers to the‘‘chemical strain’’ component of total strain owing to aperturechange. In the usual manner, total strain, e, can be spectrallydecomposed into components due to mechanical, M, chemical, CH,and thermal, T, behaviors as, � ¼ �M þ �T þ �CH . Considering onlythermal and chemical effects, the thermal/chemical strain is�TC ¼ �T þ A, where A is some constant representing the chemicalportion. At incremental equilibrium we have �TC ¼ aTDT þ A

which, upon rearranging, becomes

�TC ¼ DT aT þA

DT

� �, (46)

where A ¼ Db/s. This relationship provides a method to accom-modate chemical strain by altering the coefficient of thermalexpansion, aT, in FLAC3D at all nominal element volumes forrespective aperture changes, and, as desired, maintains a non-linear dependence on temperature. However, because the functionis undefined for temperature changes approaching zero, careshould be taken in its application. In physical systems whereaperture change, which is a strong function of the effective stressfield, is dominated by thermal stress, such as geothermal systems,such strains will be tracked appropriately, but in systems that arenearly isothermal this method will be ineffective in transferringinformation to the mechanical system (which may or may not benecessary, as isothermal systems are unlikely to experiencechemical strain to the same degree).

9.1. Chemical strain in cyclic loading

Chemical strain is defined here as thermo-chemo-mechani-cally irreversible reduction in fracture aperture that results in arelaxation of stress in the surrounding rock. As illustrated in Fig. 1,this process is proposed to be of significant importance infractured reservoirs and replicating it one of the primary goalsof THMC modeling. To examine this process, we consider the caseof a liquid saturated, high temperature and pressure fracturedmass subjected to a complete cycle of thermal loading andunloading (Fig. 7). The model is a pseudo three-dimensional mass(unit width in the z-direction, discretized in x and y) with zero-displacement boundaries and initially at a uniform temperature of

80 1C, and s0 ¼ 20.8 MPa. A high temperature (120 1C) andpressure (2 MPa above in-situ) source is placed at one end of thegeometry (x ¼ 0, y ¼ L/2) with a low pressure source (2 MPabelow in-situ) at the opposing end (x ¼ L, y ¼ L/2), allowing thethermal source to translate across the geometry with the fluidpressure gradient. After thermal breakthrough to the injectiontemperature, the temperature source is reversed to 80 1C, so thatthe mass then gradually declines to its initial temperature state.Progress is monitored at the central coordinate (x ¼ L/2, y ¼ L/2),and the results of temperature and aperture change versus stressat this location are displayed.

In the figure we present four cases incorporating differentassumptions of response, which may be compared to theconceptual representation of Fig. 1. Fig. 7A is the baseline case,with completely reversible permeability change (Eq. (34) only),and no feedback of this chemical strain on the stress field (Eq. (46)not used). Fig. 7B represents the case of complete permeabilityconstitutive treatment (Eqs. (34) and (41)) and includes feedbackon stress field (Eq. (46)). Fig. 7C maintains full permeabilityconstitutive treatment (as in 7b), but this time does not includefeedback on stress (Eq. (47) not used). Finally, Fig. 7D considerscomplete reversibility (as in 8a), but this time includes feedbackon the stress field (Eq. (47)).

The non-linear dependence of aperture on the temperature/stress field is evident, as is the non-linear dependence of stress ontemperature that results from the feedback of chemical strain onthe stress field. Two-dominant impacts on the system, hystereticin nature, are visible by comparing the initial, ambient systemwith the final, ambient system. Importantly, when the systemreturns to its initial state, there has been an irreversible reductionin the stress field as well as an irreversible decrease inpermeability. Neither of these occurrences, intuitively operativeand significant in natural systems, may be represented withoutthe inclusion of thermal, hydrologic, mechanical, and chemicalprocesses.

10. Conclusions

A coupled THMC simulator has been developed with thecapability of reproducing the undrained loading behavior of afractured rock mass. Reactive transport has been included inthe model via the equilibrium behavior of aqueous species(homogeneous reactions) and through kinetic considerations ofmineral precipitation and dissolution. From multi-continuumhydrogeologic analysis, multi-phase fluid behavior is coupledto the mechanical response in one continuum via dual-porosity poroelasticity and thermodynamically controlled fluid

ARTICLE IN PRESS

Fig. 7. Thermal loading/unloading cycle examining the effects of chemical strain. Parameters: E ¼ 13 GPa, v ¼ 0.22, aT ¼ 12�10�6/K.



compressibility. Permeability of the mass is followed with a newconstitutive relationship representing thermal loading and un-loading behavior: Closure of the fracture is controlled by thermal-elastic compaction and the dissolution of stress-concentratedasperities, while dilation occurs via thermal-hydraulic stressrelaxation. Bulk permeability is also modified by the precipita-tion/dissolution kinetics of mineral species. The explicit couplingbetween THC and M behaviors is shown to reproduce the rapidresponse of a loaded mass. Additional couplings have also beenexplored, and a subsequent paper [1] examines the strength ofcoupling between THMC mechanisms as well as the application ofthis model to an EGS scenario.

Chemical strain is accommodated by the permeability con-stitutive relationship, and its impact on the stress field of ageologic environment is illustrated. For the first time, we presentgeologic scale numerical results illustrating the conceptual modelthat thermal loading may lead to an irreversible reduction inaperture and stress, so that the in-situ system may be completelyaltered by a cycle of loading.

Acknowledgment

This work is the result of partial support from the USDepartment of Energy under Grant DOE-DE-FG36-04GO14289.This support is gratefully acknowledged.

References

[1] Taron J, Elsworth D. Thermal-hydrologic-mechanical-chemical processes inthe evolution of engineered geothermal reservoirs. Int J Rock Mech Min Sci2009; this issue, doi:10.1016/j.ijrmms.2009.01.007.

[2] Wawersik WR, Rudnicki JW. Terrestrial sequestration of CO2—an assessmentof research needs. Report from envited panelist workshop, May 1997, US DeptEnergy Geosci Res Prog, 1998.

[3] Rutqvist J, Wu Y-S, Tsang C-F, et al. A modeling approach for analysis ofcoupled multiphase fluid flow, heat transfer, and deformation in fracturedporous rock. Int J Rock Mech Min Sci 2002;39:429–42.

[4] Taron J, Elsworth D, Thompson G, et al. Mechanisms for rainfall-concurrentlava dome collapses at Soufriere Hills Volcano, 2000–2002. J Volcanol GeothRes 2007;160:195–209.

[5] Nemat-Nasser S, Keer LM, Parihar KS. Unstable growth of thermally inducedinteracting cracks in brittle solids. Int J Solids Struct 1977;14:409–30.

[6] Hunsbedt A, Kruger P, London AL. Recovery of energy from fracture-stimulated geothermal reservoirs. J Petrol Tech 1977;29:940–6.

[7] Pruess K. Enhanced geothermal systems (EGS) using CO2 as working fluid—anovel approach for generating renewable energy with simultaneous seques-tration of carbon. Geothermics 2006;35(4):351–67.

[8] Weyl PK. Pressure solution and the force of crystallization—a phenomen-ological theory. J Geophys Res 1959;64:2001–25.

[9] Paterson MS. Nonhydrostatic thermodynamics and its geologic applications.Rev Geophys Space Phys 1973;11:355–89.

[10] Revil A. Pervasive pressure-solution transfer: a poro-visco-plastic model.Geophys Res Lett 1999;26(2):255–8.

[11] Elsworth D, Goodman RE. Characterization of rock fissure hydraulicconductivity using idealized wall roughness profiles. Int J Rock Mech MinSci 1986;23(3):233–43.

[12] Barton N, Bandis S, Bakhtar K. Strength, deformation and conductivitycoupling of rock joints. Int J Rock Mech Min Sci 1985;22(3):121–40.

[13] Rose P, Xu T, Kovac KM, et al. Chemical stimulation in near-wellboregeothermal formations: silica dissolution in the presence of calcite at hightemperature and high pH. In: Proceedings of the 32nd workshop geothermalreservoir engineering, Stanford University; 2007.

[14] Xu T, Sonnenthal E, Spycher N, et al. TOUGHREACT—A simulation program fornon-isothermal multiphase reactive geochemical transport in variablysaturated geologic media: applications to geothermal injectivity and CO2

geological sequestration. Comp Geosci 2006;32:145–65.[15] Nami P, Schellschmidt R, Schindler M, et al. Chemical stimulation operations

for reservoir development of the deep crystalline HDR/EGS system at Soultz-Souz-Forets (France). In: Proceedings of the 32nd workshop geothermalreservoir engineering, Stanford University; 2007.

[16] De Boer RB. On the thermodynamics of pressure solution-interaction betweenchemical and mechanical forces. Geochem Cosmochem Acta 1977;41:249–56.

[17] Bower KM, Zyvoloski G. A numerical model for thermo-hydro-mechanicalcoupling in fractured rock. Int J Rock Mech Min Sci 1997;34(8):1201–11.

[18] Gawin D, Schrefler BA. Thermo-hydro-mechanical analysis of partiallysaturated porous materials. Eng Comput 1996;13(7):113.

[19] Taron J, Min K-B, Yasuhara H, et al. Numerical simulation of coupled thermo-hydro-chemo-mechanical processes through the linking of hydrothermal andsolid mechanics codes. In: Proceedings of the 41st US symposium on rockmechanics, Golden, Colo, June 17–21, 2006.

[20] Swenson D, Gosavi S, Hardeman B. Integration of poroelasticity into TOUGH2.In: Proceedings of the 29th international workshop geothermal reservoirengineering, Stanford University; 2004.

[21] Itasca Consulting Group Inc. FLAC3D Manual: fast Lagrangian analysis ofcontinua in 3 dimensions—version 2.0. Itasca Consulting Group Inc,Minneapolis; 1997.

[22] Settari A. Modeling of fracture and deformation processes in oil sands. In:Proceedings of the fourth UNITAR/UNDP conference on heavy crude and tarsands, Edmonton; 1988. Paper no. 43.

[23] Settari A, Mourits FM. Coupling of geomechanics and reservoir simulationmodels. Comp Methods Adv Geomech 1994:2151–8.

[24] Minkoff SE, Stone CM, Bryant S, et al. Coupled fluid flow and geomechanicaldeformation modeling. J Pet Sci Eng 2003;38:37–56.

[25] IAPWS, Industrial formulation 1997 for the thermodynamic properties ofwater and steam. IAPWS Release, IAPWS Secretariat, 1997: InternationalAssociation for the Properties of Water and Steam.

[26] Narasimhan TN, Witherspoon PA. An integrated finite difference methodfor analyzing fluid flow in porous media. Water Resour Res 1976;12(1):57–64.

[27] Wilson RK, Aifantis EC. On the theory of consolidation with double porosity.Int J Eng Sci 1982;20(9):1009–35.

[28] Khaled MY, Beskos DE, Aifantis EC. On the theory of consolidation withdouble porosity, III, A finite element formulation. Int J Numer Anal MethodsGeomech 1984;8(2):101–23.

[29] Cho TF, Plesha ME, Haimson BC. Continuum modelling of jounted porous rock.Int J Numer Anal Methods Geomech 1991;15:333–53.

[30] Elsworth D, Bai M. Flow-deformation response of dual-porosity media. JGeotech Eng 1992;118(1):107–24.

[31] Berryman JG, Wang HF. The elastic coefficients of double-porositymodels for fluid transport in jointed rock. J Geophys Res 1995;100(12):24,611–27.

[32] Jaeger JC, Cook NGW, Zimmerman RW. Fundamentals of rock mechanics. 4thed. Oxford: Wiley-Blackwell; 2007.

[33] Xu T, Pruess K. Modeling multiphase non-isothermal fluid flow and reactivegeochemical transport in variably saturated fractured rocks: 1. Methodology.Am J Sci 2001;301:16–33.

[34] Steefel CI, Lasaga AC. A coupled model for transport of multiple chemicalspecies and kinetic precipitation/dissolution reactions with applications toreactive flow in single phase hydrothermal system. Am J Sci 1994;294:529–92.

[35] Lasaga AC. Chemical kinetics of water-rock interactions. J Geophys Res 1984;89(B6):4009–25.

[36] Lasaga AC, Soler JM, Ganor J, et al. Chemical weathering rate laws and globalgeochemical cycles. Geochem Cosmochem Acta 1994;58:2361–86.

[37] Lasaga AC. Rate laws of chemical reactions. In: Lasaga AC, Kirkpatrick RJ,editors. Kinetics of geochemical processes, reviews in mineralogy, vol. 8.Washington: Mineral Soc Amer; 1981.

[38] Carroll S, Mroczek E, Alai M, et al. Amorphous silica precipitation (60–120 1C):comparison of laboratory and field rates. Geochem Cosmochem Acta1998;62:1379–96.

[39] Rimstidt JD, Barnes HL. The kinetics of silica-water reactions. GeochemCosmochem Acta 1980;44:1683–99.

[40] Xu T, Sonnenthal E, Spycher N, et al. TOUGHREACT user’s guide: a simulationprogram for non-isothermal multiphase reactive geochemical transport invariably saturated geologic media. Report LBNL-55460, Lawrence BerkeleyNational Laboratory, 2004.

[41] Helgeson HC, Kirkham DH, Flowers DC. Theoretical prediction of thethermodynamic behavior of aqueous electrolytes at high pressures andtemperatures: IV. Calculation of activity coefficients, osmotic coefficients, andapparent molal and standard and relative partial molal properties to 600 1Cand 5 kb. Am J Sci 1981;281:1249–516.

[42] Warren JE, Root PJ. The behavior of naturally fractured reservoirs. Soc PetrolEng J 1963;3:245–55.

[43] Barenblatt GI, Zheltov YP, Kochina IN. Basic concepts in the theory of seepageof homogeneous liquids in fissured rocks. J Appl Math Mech 1960;24:1286–303.

[44] Pruess K, Narasimhan TN. On fluid reserves and the production of superheatedsteam from fractured, vapor-dominated geothermal reservoirs. J Geophys Res1982;87(B11):9329–39.

[45] Pruess K, Narasimhan TN. A practical method for modeling fluid and heatflow in fractured porous media. Soc Petrol Eng J 1985;25(1):14–26.

[46] Bai M, Elsworth D, Roegiers J-C. Modeling of naturally fractured reservoirsusing deformation dependent flow mechanism. Int J Rock Mech Min Sci1993;30(7):1185–91.

[47] Biot MA. Theory of propagation of elastic waves in a fluid-saturated poroussolid, II. Higher frequency range. J Acoust Soc Am 1956;28(2):179–91.

[48] Biot MA. General theory of three-dimensional consolidation. J Appl Phys1941;12:155–64.

[49] Biot MA. Theory of propagation of elastic waves in a fluid-saturated poroussolid, I. Low-frequency range. J Acoust Soc Am 1956;28(2):168–78.

[50] Biot MA. Mechanics of deformation and acoustic propagation in porousmedia. J Appl Phys 1962;33(4):1482–98.

ARTICLE IN PRESS



[51] Berryman JG, Pride SR. Models for computing geomechanical constants ofdouble-porosity materials from the constituents’ properties. J Geophys Res2002;107(B3):2052.

[52] Biot MA, Willis DG. The elastic coefficients of the theory of consolidation.J Appl Mech 1957;24:594–601.

[53] Zimmerman RW, Chen G, Hadgu T, Bodvarsson GS. A numerical dual-porositymodel with semi-analytical treatment of fracture/matrix flow. Water ResourRes 1993;29:2127–37.

[54] Detournay E, Cheng AH-D. Fundamentals of poroelasticity. In: Hudson JA, editor.Comprehensive rock engineering. New York: Pergamon; 1993. p. 113–71.

[55] Mandel J. Consolidation des sols (etude mathematique). Geotechnique1953;3:287–99.

[56] Abousleiman Y, Cheng AH-D, Cui L, et al. Mandel’s problem revisited.Geotechnique 1996;46(2):187–95.

[57] Rice JR, Cleary MP. Some basic stress diffusion solutions for fluid-saturatedelastic porous media with compressible constituents. Rev Geophys SpacePhys 1976;14(2):227–40.

[58] Wang HF. Theory of linear poroelasticity. Princeton: Princeton UniversityPress; 2000.

[59] Carslaw HS, Jaeger JC. Conduction of heat in solids. 2nd ed. Oxford: OxfordUniversity Press; 1959.

[60] Geertsma J. Problems of rock mechanics in petroleum production engineer-ing. In: Proceedings of the first international congress rock mechanics, vol. 1,Lisbon, 1966. p. 585–94.

[61] Min K-B, Rutqvist J, Elsworth D. Chemically and mechanically mediatedinfluences on the transport and mechanical characteristics of rock fractures.Int J Rock Mech Min Sci 2009;46(1):80–9.

[62] Stephenson LP, Plumley WJ, Palciauskas VV. A model for sandstonecompaction by grain interpenetration. J Sediment Petrol 1992;62:11–22.

[63] Yasuhara H, Elsworth D, Polak A. A mechanistic model for compaction ofgranular aggregates moderated by pressure solution. J Geophys Res 2003;108(B11).

[64] Snow DT. Anisotropic permeability of fractured media. Water Resour Res1969;5(6):1273–89.

[65] Witherspoon PA, Wang JSY, Iwai K, et al. Validity of cubic law for fluid flow ina deformable rock fracture. Water Resour Res 1980;16(6):1016–24.

[66] Hardin EL, Barton N, Lingle R, Board MP, Voegele MD. A heated flatjack testseries to measure the thermomechanical and transport properties of in siturock masses. Report ONWI-260, Columbus, Ohio: Office of Nuclear WasteIsolation; 1982.

[67] Dunn EJ. Reports on the Bendigo Goldfield. Special reports nos. 1 and 2,Department of Mines, Victoria, Australia, 1896. p. 16.

[68] Taber S. The mechanics of vein formation (with discussion). Am Inst Min MetEng Trans 1920;61:3–41.

[69] Wiltschko DV, Morse JW. Crystallization pressure versus ‘‘crack seal’’ as themechanism for banded veins. Geology 2001;29(1):79–82.

[70] Dewers T, Ortoleva P. Force of crystallization during the growth of siliceousconcretions. Geology 1990;18:204–7.

[71] Maliva RG, Siever R. Diagenetic replacement controlled by force ofcrytallization. Geology 1988;16:688–91.

[72] Lichtner PC. Continuum formulation of multicomponent-multiphase reactivetransport. In: Lichtner PC, Steefel CI, Oelkers EH, editors. Reactive transport inporous media. Washington, DC: Mineral Soc Amer; 1996.

[73] Bear J. Dynamics of fluids in porous media. New York: Elsevier; 1972.

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Author's personal copy - ems.psu.eduelsworth/publications/journals/2009_j... · Author's personal copy ... thermal-hydro-mechanical asperity compaction/dilation, ... by fracture properties